Math Counts Test Review 2008 State. Total angle = 6*180 = 1080 3. What is the greatest number of interior right angles a convex octagon can have? Let.

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Presentation on theme: "Math Counts Test Review 2008 State. Total angle = 6*180 = 1080 3. What is the greatest number of interior right angles a convex octagon can have? Let."— Presentation transcript:

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Math Counts Test Review 2008 State

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Total angle = 6*180 = What is the greatest number of interior right angles a convex octagon can have? Let X be number of right angles, all other angles must be less than 180: 90 X + (8 – X) * 180 > X < 8*180 – X < 360 X < 4 Answer: 3

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X / (15 – X) = 2/7 21. A circle with a radius of 2 units has its center at (0, 0). A circle with a radius of 7 units has its center at (15, 0). A line tangent to both circles intersects the x- axis at (x, 0) to the right of the origin. What is the value of x? Express your answer as a common fraction. 7 X = 2 (15 – X) = 30 – 2X 9 X = 30 X = 10/3 x A B C(15,0) O Note that  OAX   CBX

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For one round of loop, we can find: 23. A circular cylindrical post with a circumference of 4 feet has a string wrapped around it, spiraling from the bottom of the post to the top of the post. The string evenly loops around the post exactly four full times, starting at the bottom edge and finishing at the top edge. The height of the post is 12 feet. What is the length, in feet, of the string? 4 3 length of string = 5 For the 4 loops, The string length = 5*4 = 20

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From any a1, a2, a3, a4, we can generate the sequence: 26. Kevin will start with the integers 1, 2, 3 and 4 each used exactly once and written in a row in any order. Then he will find the sum of the adjacent pairs of integers in each row to make a new row, until one integer is left. For example, if he starts with 3, 2, 1, 4, and then takes sums to get 5, 3, 5, followed by 8, 8, he ends with the final sum 16. Including all of Kevin’s possible starting arrangements of the integers 1, 2, 3 and 4, how many possible final sums are there? a1+a2, a2+a3, a3+a4; a1+a2+a2+a3, a2+a3+a3+a4; a1+3a2+3a3+a4; From 1, 2, 3, 4, there are 6 ways to pick different a1 & a4 Hence the maximum # of possible sums are 6. Note that 1, 2, 3, 4 and 2, 1, 4, 3 generate the same sum We have the final answer: 6 – 1 = 5

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Let X be total number of members 27. A rectangular band formation is a formation with m band members in each of r rows, where m and r are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by 1 and reduces the number of rows by 2, there are exactly enough places in the new formation for each band member. What is the largest number of members the band could have? X = r m (1) X = (m + 1) *(r -2) = r m + r – 2m – (2) (2) – (1): r – 2m – 4 = 0  r = 2m (3) (3) to (2): X = (m+1)(2m+4-2)=2(m+1) (4) Since X < 100, (m+1) 2 = X/2 < 50 Largest possible integer m is 7. Answer: X = 98

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Note that the height of the cylinder is 4*2 = A right cylinder with a base radius of 3 units is inscribed in a sphere of radius 5 units. What is the total volume, in cubic units, of the space inside the sphere and outside the cylinder? Express your answer as a common fraction in terms of . 5 o 3 4 V c = (3)2  * 8 = 72  V s = 4/3 * 5 3  = 500/3  Answer = 500/3  – 72  = 284/3 

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It’s easier to find the reverse possibility here. 30. An o-Pod MP3 player stores and plays entire songs. Celeste has 10 songs stored on her o-Pod. The time length of each song is different. When the songs are ordered by length, the shortest song is only 30 seconds long and each subsequent song is 30 seconds longer than the previous song. Her favorite song is 3 minutes, 30 seconds long. The o- Pod will play all the songs in random order before repeating any song. What is the probability that she hears the first 4 minutes, 30 seconds of music - there are no pauses between songs - without hearing every second of her favorite song? Express your answer as a common fraction. That is, to hear the entire favorite song. That leaves 4m30sec – 3min30sec = 1 min to hear other songs. She can only hear (1) her favorite song first or (2) the first song first, and then her favorite song or (3) the second song first, and then her favorite song Prob 1 = 1/10; Prob 2 = 1/10 * 1/9; Prob 3 = 1/10 * 1/9 Answer = 1 – 1/10 – 1/90 – 1/90 = 9/10 – 2/90 = 79/90

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Let X be the length of the jet M3. At a particular speed, a jumbo jet can travel its own length in 20 seconds. At this same speed, the jumbo jet taxied completely past a 710-foot-long hangar in 70 seconds, as shown. What is the length of the jumbo jet? (X + 710) / 70 = X/20 = speed of the jet 70 X = 20X * X = 710 * 20 X = 71 * 20 / 5 = 284

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Let H and h be the heights of two candles M7. One night two cylindrical wax candles of different heights and different diameters were lit. One of the candles was 20 cm taller than the other. They were both lit at the same time and each burned at a steady rate. Five hours after they were lit they were both the same height. The taller one burned all of its wax six hours after it was lit, and the shorter one burned all of its wax 10 hours after it was lit. What was the ratio of the original height of the shorter candle to the original height of the taller candle? Express your answer as a common fraction. Let R and r be their speed of burning (in/hr) H = 6 * R h = 10 r H – 5 R = h – 5 r H – 5 * H/6 = h – 5 * h/10 1/6 H = ½ h  h/H = 1/3

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Let a, b, c be the lengths of any obtuse triangle, with a < c & b < c. For the triangle to be obtuse, we have: M8. Each of three, standard, six-sided dice is rolled once. What is the probability that the three numbers rolled are the lengths of the sides of an obtuse triangle? Express your answer as a common fraction. a 2 + b 2 < c 2 and c < a + b List of triplets meeting the criteria include: 1,1,(3,4,5,6); 1,2,(3,4,5,6); 1,3,(4,5,6); 1,4,(5,6); 1,5,6; 1,6,(none) 2,2,(3,4,5,6);2,3,(4,5,6); 2,4,(5,6); 2,5,6; 2,6,(none) 3,1,(4,5,6);3,2,(4,5,6); 3,3,(5,6); 3,4,6; 3,5,6; 3,6,(none) Total # of triplets: 6*6*6 # of triplets fitting the requirement: 13 * 3 = 39 (since c can be any of the thee dices) Answer: 39/(6*6*6) = 13/72 4,1,(5,6);4,2,(5,6); 4,3,6; 4,4,6; 4,(5,6),(none) 5,1,6; 5,2,6; 5,3,6; 5,(4,5,6),(none); 6,(1,2,3,4,5,6), (none)

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M10. A circular tabletop is divided into four congruent sectors by two diameters that are perpendicular to each other. Each sector is to be painted with one of four colors. How many distinct ways can the table be painted? (A color may be used on more than one sector, but paintings that are the same after a rotation are not considered distinct.) 2) # of ways with 2 different colors: 6 * 4 = 24 Reason: # of ways to pick 2colors: 4*3/2 = 6 For any pick of 2 colors, there are 4 ways: ABAB, AABB,AAAB, ABBB 1) # of ways with 1 color: 4 (Reason: 4 colors to choose, each with 1 painting) 3) # of ways with 3 different colors: 6 * 6 = 36 Reason: # of ways to pick 3 colors (= # of ways to drop 1 color): 4 For each pick of 3 colors, there are 9 ways to paint: 4) # of ways with 4 different colors: 6 Reason: # of ways to color without rotation: 4! = 24 Each way can be repeated 4 times with the rotation, hence 24/4 = 6 Total ways of painting = 1) + 2) + 3) + 4) = = 70 AABC, AACB, BBAC, BBCA, CCAB, CCBA, ABAC, BABC, CACB

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Advanced Math Practice

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A. In rectangle ABCD, AB = 12 and BC = 10. Points E and F lie inside rectangle so that BE = 9, DF = 8, BE // DF, EF // AB, and line BE intersects segment AD. The length EF can be expressed in the form m  n - p, where m, n, and p are positive integers and n is not divisible by the square of any prime. Find m + n + p. B. Suppose that a parabola has vertex (1/4, -9/8) and equation Y = aX 2 + bX + c, where a > 0 and a + b + c is an integer. The minimum possible value of a can be written in the form p/q, where p and q are relatively prime positive integers. Find p + q.